<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c7">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  st not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c7">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b2">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b2">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> holds <br/><font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">P</font></span><br/><b>let </b><font color="Maroon" title="c8">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  st not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c7">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Olive" title="b1">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> holds <br/><font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">P</font></span><br/><b>let </b><font color="Maroon" title="c9">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c7">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c9">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c9">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> implies <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">P</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:105"/><i><font color="Green" title="E79">A1</font></i>: 
not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c7">C</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E2:105"/><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon" title="c9">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font>
 <b>and </b><br/><a NAME="E3:105"/><i><font color="Green" title="E81">A3</font></i>: 
<font color="Maroon" title="c9">U</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:105"/><i><font color="Green" title="E82">A4</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:105"/><i><font color="Green" title="E83">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">T2C</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2 ;<br/>
<a NAME="E7:105"/><i><font color="Green" title="E84">A6</font></i>: 
<font color="Maroon" title="c10">T2C</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T89" title="JORDAN2C:th.89">JORDAN2C:89</a></i>;<br/>
<a NAME="E8:105"/><b>then </b>
<font color="Maroon" title="c9">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="connsp_2.html#T21" title="CONNSP_2:th.21">CONNSP_2:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">P</font> = <font color="Maroon" title="c8">P</font> as  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></i>;<br/>
<a NAME="E10:105"/><i><font color="Green" title="E85">A7</font></i>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c11">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">Z</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c13">V</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> ;<br/>
<a NAME="E11:105"/><i><font color="Green" title="E86">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<a NAME="E1:105_1"/><i><font color="Green" title="E87">A9</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span>  or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:105_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c15">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:105_1_1"/><i><font color="Green" title="E88">A10</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">A</font>
 <b>and </b><br/><a NAME="E4:105_1_1"/><i><font color="Green" title="E89">A11</font></i>: 
<font color="Maroon" title="c15">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:105_1_1"/>
 ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">A</font> &amp; <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> )
 
<b>by </b><i><a class="txt" href="#E4:105_1_1"><i><font color="Green" title="E89">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:105_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="#E5:105"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="#E3:105_1_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:105_1"/>
<font color="Maroon" title="c9">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:105"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:105_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:105_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E4:105_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font></span><br/>

<b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font> )</span><br/>

<b>assume </b><a NAME="E5:105_1"/><i><font color="Green" title="E88">A12</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> or  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_1"><b>suppose </b></a><a NAME="E1:105_1_2_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font></span><br/><div class="add"><b>hence </b><a NAME="E2:105_1_2_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2"><b>suppose </b></a><a NAME="E1:105_1_2_2"/>
not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c14">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E5:105"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="#E5:105_1"><i><font color="Green" title="E88">A12</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/><a NAME="E3:105_1_2_2"/><i><font color="Green" title="E89">A13</font></i>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T40" title="CONNSP_1:th.40">CONNSP_1:40</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_2_2_1"/>
(  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">U</font> or  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:105_1_2_2_1_1"/>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font></span><br/><div class="add"><a NAME="E2:105_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font>
 <b>by </b><i><a class="txt" href="#E3:105_1_2_2"><i><font color="Green" title="E89">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:105_1_2_2_1_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:105_1_2_2_1_2"/><i><font color="Green" title="E90">A14</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font></span><br/><div class="add"><a NAME="E2:105_1_2_2_1_2"/>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T43" title="CONNSP_1:th.43">CONNSP_1:43</a></i>;<br/><a NAME="E3:105_1_2_2_1_2"/><b>then </b>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:105_1_2_2_1_2"><i><font color="Green" title="E90">A14</font></i></a></i>;<br/><a NAME="E4:105_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:105_1_2_2"><i><font color="Green" title="E89">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E5:105_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:105_1_2_2_1_2"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:105"/><i><font color="Green" title="E87">A15</font></i>: 
<font color="Maroon" title="c11">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E13:105"/>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c11">P</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> )</span><br/>

<b>assume </b><a NAME="E1:105_2"/><i><font color="Green" title="E88">A16</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c11">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font></span><br/>

<a NAME="E2:105_2"/><b>then </b><i><font color="Green" title="E89">A17</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">P</font> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="#E10:105"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>assume </b><a NAME="E3:105_2"/>
not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:105_2"/><b>then </b>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">U</font>
 
<b>by </b><i><a class="txt" href="#E11:105"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="#E1:105_2"><i><font color="Green" title="E88">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:105_2"/><b>then </b>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span>  or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">U</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">O</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:105_2"/><i><font color="Green" title="E90">A18</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">O</font>
 <b>and </b><br/><a NAME="E8:105_2"/><i><font color="Green" title="E91">A19</font></i>: 
<font color="Maroon" title="c15">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> : ( <font color="Olive" title="b1">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E12:105"><i><font color="Green" title="E87">A15</font></i></a>, <a class="txt" href="#E2:105_2"><i><font color="Green" title="E89">A17</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:105_2"/><i><font color="Green" title="E92">A20</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">O</font>
 <b>and </b><br/><a NAME="E11:105_2"/><i><font color="Green" title="E93">A21</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E12:105_2"/><i><font color="Green" title="E94">A22</font></i>: 
<font color="Maroon" title="c16">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">U</font>
 <b>by </b><i><a class="txt" href="#E8:105_2"><i><font color="Green" title="E91">A19</font></i></a></i>;<br/>
<a NAME="E13:105_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">X</font>
 
<b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E11:105_2"><i><font color="Green" title="E93">A21</font></i></a>, <a class="txt" href="#E12:105_2"><i><font color="Green" title="E94">A22</font></i></a>, <a class="ref" href="jordan.html#T68" target="_self" title="JORDAN:th.68">Th68</a></i>;<br/>
<b>hence </b><a NAME="E14:105_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:105_2"><i><font color="Green" title="E89">A17</font></i></a>, <a class="txt" href="#E7:105_2"><i><font color="Green" title="E90">A18</font></i></a>, <a class="txt" href="#E10:105_2"><i><font color="Green" title="E92">A20</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:105"/><b>then </b>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c11">P</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="#E4:105"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">p1</font>, <font color="Maroon" title="c15">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon" title="c16">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E16:105"/><i><font color="Green" title="E88">A23</font></i>: 
<font color="Maroon" title="c16">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font>
 <b>and </b><br/><a NAME="E17:105"/><i><font color="Green" title="E89">A24</font></i>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">A</font>
 <b>and </b><br/><a NAME="E18:105"/><i><font color="Green" title="E90">A25</font></i>: 
<font color="Maroon" title="c16">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="ref" href="borsuk_4.html#T84" title="BORSUK_4:th.84">BORSUK_4:84</a></i>;<br/>
<a NAME="E19:105"/><i><font color="Green" title="E91">A26</font></i>: 
<font color="Maroon" title="c9">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="connsp_1.html#T34" title="CONNSP_1:th.34">CONNSP_1:34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_3"/>
( <font color="Maroon" title="c11">P</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> or  not <font color="Maroon" title="c11">P</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1"><b>suppose </b></a><a NAME="E1:105_3_1"/>
<font color="Maroon" title="c11">P</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c17">P</font> = <font color="Maroon" title="c11">P</font> as  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><b>consider </b><font color="Maroon" title="c18">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:105_3_1"/><i><font color="Green" title="E92">A27</font></i>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">U</font>
 <b>by </b><i><a class="txt" href="#E19:105"><i><font color="Green" title="E91">A26</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c18">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E4:105_3_1"><i><font color="Green" title="E92">A27</font></i></a></i>;<br/><a NAME="E6:105_3_1"/><i><font color="Green" title="E93">A28</font></i>: 
<font color="Maroon" title="c17">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T76" title="TOPREAL6:th.76">TOPREAL6:76</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">PC</font> = <font color="Maroon" title="c17">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font> as  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">r</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E9:105_3_1"/><i><font color="Green" title="E94">A29</font></i>: 
<font color="Maroon" title="c20">PC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="#E6:105_3_1"><i><font color="Green" title="E93">A28</font></i></a>, <a class="ref" href="jordan.html#T26" target="_self" title="JORDAN:th.26">Th26</a></i>;<br/><a NAME="E10:105_3_1"/>
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">PC</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E11:105_3_1"/><b>then </b><i><font color="Green" title="E95">A30</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="#E9:105_3_1"><i><font color="Green" title="E94">A29</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c22">D</font> =  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>;<br/><b>set </b><font color="Maroon" title="c23">S</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>;<br/><b>set </b><font color="Maroon" title="c24">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E12:105_3_1"/><i><font color="Green" title="E96">A31</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></i>;<br/><a NAME="E13:105_3_1"/><i><font color="Green" title="E97">A32</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="brouwer.html#T3" title="BROUWER:th.3">BROUWER:3</a></i>;<br/><a NAME="E14:105_3_1"/><i><font color="Green" title="E98">A33</font></i>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></i>;<br/><a NAME="E15:105_3_1"/><i><font color="Green" title="E99">A34</font></i>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19" title="TOPREAL9:th.19">TOPREAL9:19</a></i>;<br/><a NAME="E16:105_3_1"/><i><font color="Green" title="E100">A35</font></i>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T16" title="TOPREAL9:th.16">TOPREAL9:16</a></i>;<br/><a NAME="E17:105_3_1"/>
<font color="Maroon" title="c16">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="#E18:105"><i><font color="Green" title="E90">A25</font></i></a>, <a class="txt" href="#E11:105_3_1"><i><font color="Green" title="E95">A30</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E18:105_3_1"/><b>then </b>
<font color="Maroon" title="c16">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:105_3_1"><i><font color="Green" title="E97">A32</font></i></a>, <a class="txt" href="#E16:105_3_1"><i><font color="Green" title="E100">A35</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">R</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">A</font></span>)</span><b> such that </b><br/><a NAME="E20:105_3_1"/><i><font color="Green" title="E101">A36</font></i>: 
<font color="Maroon" title="c25">R</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E21:105_3_1"/><i><font color="Green" title="E102">A37</font></i>: 
<font color="Maroon" title="c25">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c16">A</font>
 <b>by </b><i><a class="txt" href="#E16:105"><i><font color="Green" title="E88">A23</font></i></a>, <a class="txt" href="jordan.html#E89"><i><font color="Green" title="E75">Lm14</font></i></a></i>;<br/><a NAME="E22:105_3_1"/>
<font color="Maroon" title="c17">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">PC</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E23:105_3_1"/><b>then </b><i><font color="Green" title="E103">A38</font></i>: 
<font color="Maroon" title="c17">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="#E9:105_3_1"><i><font color="Green" title="E94">A29</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c19">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E25:105_3_1"/><i><font color="Green" title="E104">A39</font></i>: 
<font color="Maroon" title="c26">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E26:105_3_1"/><i><font color="Green" title="E105">A40</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c17">P</font> implies <font color="Maroon" title="c26">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <font color="Maroon" title="c26">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E10:105"><i><font color="Green" title="E85">A7</font></i></a>, <a class="txt" href="#E16:105"><i><font color="Green" title="E88">A23</font></i></a>, <a class="txt" href="#E17:105"><i><font color="Green" title="E89">A24</font></i></a>, <a class="txt" href="#E18:105"><i><font color="Green" title="E90">A25</font></i></a>, <a class="txt" href="#E4:105_3_1"><i><font color="Green" title="E92">A27</font></i></a>, <a class="txt" href="#E11:105_3_1"><i><font color="Green" title="E95">A30</font></i></a>, <a class="txt" href="#E20:105_3_1"><i><font color="Green" title="E101">A36</font></i></a>, <a class="txt" href="#E21:105_3_1"><i><font color="Green" title="E102">A37</font></i></a>, <a class="txt" href="jordan.html#E90"><i><font color="Green" title="E76">Lm15</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c27">g</font> =  <a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font>;<br/><b>set </b><font color="Maroon" title="c28">h</font> =  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font>;<br/><a NAME="E27:105_3_1"/><i><font color="Green" title="E106">A41</font></i>: 
 <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="#E12:105_3_1"><i><font color="Green" title="E96">A31</font></i></a>, <a class="txt" href="#E13:105_3_1"><i><font color="Green" title="E97">A32</font></i></a>, <a class="txt" href="#E14:105_3_1"><i><font color="Green" title="E98">A33</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">F</font> = <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c26">f</font></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <b>by </b><i><a class="txt" href="#E12:105_3_1"><i><font color="Green" title="E96">A31</font></i></a>, <a class="txt" href="#E13:105_3_1"><i><font color="Green" title="E97">A32</font></i></a>, <a class="txt" href="#E14:105_3_1"><i><font color="Green" title="E98">A33</font></i></a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/><a NAME="E29:105_3_1"/>
<font color="Maroon" title="c19">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T17" target="_self" title="JORDAN:th.17">Th17</a></i>;<br/><a NAME="E30:105_3_1"/><b>then </b>
 <a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T64" target="_self" title="JORDAN:th.64">Th64</a></i>;<br/><a NAME="E31:105_3_1"/><b>then </b><i><font color="Green" title="E107">A42</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c26">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E25:105_3_1"><i><font color="Green" title="E104">A39</font></i></a></i>;<br/><a NAME="E32:105_3_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c19">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="toprns_1.html#T29" title="TOPRNS_1:th.29">TOPRNS_1:29</a></i>;<br/><a NAME="E33:105_3_1"/><b>then </b><i><font color="Green" title="E108">A43</font></i>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/><a NAME="E34:105_3_1"/><b>then </b>
 <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T66" target="_self" title="JORDAN:th.66">Th66</a></i>;<br/><a NAME="E35:105_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c26">f</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E31:105_3_1"><i><font color="Green" title="E107">A42</font></i></a></i>;<br/><a NAME="E36:105_3_1"/><b>then </b><i><font color="Green" title="E109">A44</font></i>: 
<font color="Maroon" title="c29">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E27:105_3_1"><i><font color="Green" title="E106">A41</font></i></a>, <a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c29">F</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_3_1_1_1"/>
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">F</font> or  not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_3_1_1_1_1"/><i><font color="Green" title="E110">A45</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c29">F</font></span><br/><div class="add"><a NAME="E2:105_3_1_1_1_1"/><i><font color="Green" title="E111">A46</font></i>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T18" title="TOPREAL9:th.18">TOPREAL9:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font> or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font> )
 </span><b>by </b><i><a class="txt" href="#E13:105_3_1"><i><font color="Green" title="E97">A32</font></i></a>, <a class="txt" href="#E1:105_3_1_1_1_1"><i><font color="Green" title="E110">A45</font></i></a>, <a class="txt" href="#E2:105_3_1_1_1_1"><i><font color="Green" title="E111">A46</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_3_1_1_1_1_1_1_1"/><i><font color="Green" title="E112">A47</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c29">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font></span><br/><div class="add"><a NAME="E2:105_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c29">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:105_3_1_1_1_1"><i><font color="Green" title="E110">A45</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:105_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c29">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="txt" href="#E12:105_3_1"><i><font color="Green" title="E96">A31</font></i></a>, <a class="txt" href="#E15:105_3_1"><i><font color="Green" title="E99">A34</font></i></a>, <a class="txt" href="#E1:105_3_1_1_1_1_1_1_1"><i><font color="Green" title="E112">A47</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_3_1_1_1_1_1_1_2"/><i><font color="Green" title="E112">A48</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c29">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font></span><br/><div class="add"><a NAME="E2:105_3_1_1_1_1_1_1_2"/><i><font color="Green" title="E113">A49</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:105_3_1_1_1_1_1_1_2"/>
not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P</font>
 <b>by </b><i><a class="txt" href="#E15:105_3_1"><i><font color="Green" title="E99">A34</font></i></a>, <a class="txt" href="#E23:105_3_1"><i><font color="Green" title="E103">A38</font></i></a>, <a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:105_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E114">A50</font></i>: 
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/><a NAME="E5:105_3_1_1_1_1_1_1_2"/><i><font color="Green" title="E115">A51</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E33:105_3_1"><i><font color="Green" title="E108">A43</font></i></a>, <a class="ref" href="jordan.html#T65" target="_self" title="JORDAN:th.65">Th65</a></i>;<br/><a NAME="E6:105_3_1_1_1_1_1_1_2"/>
 <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 <b>by </b><i><a class="txt" href="#E33:105_3_1"><i><font color="Green" title="E108">A43</font></i></a>, <a class="ref" href="jordan.html#T67" target="_self" title="JORDAN:th.67">Th67</a></i>;<br/><a NAME="E7:105_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E116">A52</font></i>: 
not <font color="Maroon" title="c30">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font>
 <b>by </b><i><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/><a NAME="E8:105_3_1_1_1_1_1_1_2"/><i><font color="Green" title="E117">A53</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon" title="c29">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> = 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c26">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:105_3_1_1_1_1"><i><font color="Green" title="E110">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E13:105_3_1"><i><font color="Green" title="E97">A32</font></i></a>, <a class="txt" href="#E14:105_3_1"><i><font color="Green" title="E98">A33</font></i></a>, <a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="txt" href="#E2:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E113">A49</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:105_3_1"><i><font color="Green" title="E97">A32</font></i></a>, <a class="txt" href="#E14:105_3_1"><i><font color="Green" title="E98">A33</font></i></a>, <a class="txt" href="#E26:105_3_1"><i><font color="Green" title="E105">A40</font></i></a>, <a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="txt" href="#E4:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E114">A50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="txt" href="#E5:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E115">A51</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c19">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font>
<b>by </b><i><a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E10:105_3_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c29">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c30">x</font>
 <b>by </b><i><a class="txt" href="#E12:105_3_1"><i><font color="Green" title="E96">A31</font></i></a>, <a class="txt" href="#E1:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A48</font></i></a>, <a class="txt" href="#E7:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E116">A52</font></i></a>, <a class="txt" href="#E8:105_3_1_1_1_1_1_1_2"><i><font color="Green" title="E117">A53</font></i></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:105_3_1_1_1_1"/>
not <font color="Maroon" title="c30">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c29">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_3_1_1_1_2"/>
not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c29">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c29">F</font></span><br/><div class="add"><b>hence </b><a NAME="E2:105_3_1_1_1_2"/>
not <font color="Maroon" title="c30">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c29">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E38:105_3_1"/><b>then </b>
not <font color="Maroon" title="c29">F</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> 
 <b>by </b><i><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/><b>hence </b><a NAME="E39:105_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E36:105_3_1"><i><font color="Green" title="E109">A44</font></i></a>, <a class="ref" href="brouwer.html#T14" title="BROUWER:th.14">BROUWER:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2"><b>suppose </b></a><a NAME="E1:105_3_2"/><i><font color="Green" title="E92">A54</font></i>: 
not <font color="Maroon" title="c11">P</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c17">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:105_3_2"/><i><font color="Green" title="E93">A55</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="#E1:105"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:105_3_2"/><i><font color="Green" title="E94">A56</font></i>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Z</font>
 <b>and </b><br/><a NAME="E6:105_3_2"/><i><font color="Green" title="E95">A57</font></i>: 
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c7">C</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:105_3_2"><i><font color="Green" title="E93">A55</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">P1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:105_3_2"/><i><font color="Green" title="E96">A58</font></i>: 
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P1</font>
 <b>and </b><br/><a NAME="E9:105_3_2"/><i><font color="Green" title="E97">A59</font></i>: 
<font color="Maroon" title="c19">P1</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="#E6:105_3_2"><i><font color="Green" title="E95">A57</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c20">U1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:105_3_2"/><i><font color="Green" title="E98">A60</font></i>: 
( <font color="Maroon" title="c20">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">P1</font> &amp; <font color="Maroon" title="c20">U1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <b>and </b><br/><a NAME="E12:105_3_2"/>
<font color="Maroon" title="c20">U1</font> is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="#E9:105_3_2"><i><font color="Green" title="E97">A59</font></i></a>, <a class="ref" href="jordan2c.html#T17" title="JORDAN2C:th.17">JORDAN2C:17</a></i>;<br/><a NAME="E13:105_3_2"/>
<font color="Maroon" title="c20">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E7:105"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="#E11:105_3_2"><i><font color="Green" title="E98">A60</font></i></a>, <a class="ref" href="connsp_2.html#T21" title="CONNSP_2:th.21">CONNSP_2:21</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">P1</font> = <font color="Maroon" title="c19">P1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E5:105_3_2"><i><font color="Green" title="E94">A56</font></i></a>, <a class="txt" href="#E8:105_3_2"><i><font color="Green" title="E96">A58</font></i></a>, <a class="txt" href="#E9:105_3_2"><i><font color="Green" title="E97">A59</font></i></a>, <a class="txt" href="#E11:105_3_2"><i><font color="Green" title="E98">A60</font></i></a>, <a class="ref" href="jordan2c.html#D3" title="JORDAN2C:def.3">JORDAN2C:def 3</a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">p</font> = <font color="Maroon" title="c17">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E5:105_3_2"><i><font color="Green" title="E94">A56</font></i></a>, <a class="txt" href="#E8:105_3_2"><i><font color="Green" title="E96">A58</font></i></a></i>;<br/><a NAME="E16:105_3_2"/><i><font color="Green" title="E99">A61</font></i>: 
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">P1</font>
 <b>by </b><i><a class="txt" href="#E5:105_3_2"><i><font color="Green" title="E94">A56</font></i></a>, <a class="txt" href="#E8:105_3_2"><i><font color="Green" title="E96">A58</font></i></a></i>;<br/><a NAME="E17:105_3_2"/><i><font color="Green" title="E100">A62</font></i>: 
<font color="Maroon" title="c21">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T76" title="TOPREAL6:th.76">TOPREAL6:76</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c23">PC</font> = <font color="Maroon" title="c21">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font> as  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c24">rv</font> being  <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E20:105_3_2"/><i><font color="Green" title="E101">A63</font></i>: 
<font color="Maroon" title="c23">PC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c24">rv</font>
 <b>by </b><i><a class="txt" href="#E17:105_3_2"><i><font color="Green" title="E100">A62</font></i></a>, <a class="ref" href="jordan.html#T26" target="_self" title="JORDAN:th.26">Th26</a></i>;<br/><a NAME="E21:105_3_2"/>
not <font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c24">rv</font>
 <b>by </b><i><a class="txt" href="#E1:105_3_2"><i><font color="Green" title="E92">A54</font></i></a>, <a class="ref" href="jordan2c.html#T16" title="JORDAN2C:th.16">JORDAN2C:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E23:105_3_2"/><i><font color="Green" title="E102">A64</font></i>: 
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font>
 <b>and </b><br/><a NAME="E24:105_3_2"/><i><font color="Green" title="E103">A65</font></i>: 
not <font color="Maroon" title="c25">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c24">rv</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">u</font> = <font color="Maroon" title="c25">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E23:105_3_2"><i><font color="Green" title="E102">A64</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c27">r</font> = <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c22">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>;<br/><a NAME="E26:105_3_2"/>
<font color="Maroon" title="c11">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c21">P1</font>
 <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E1:105_3_2"><i><font color="Green" title="E92">A54</font></i></a>, <a class="txt" href="#E11:105_3_2"><i><font color="Green" title="E98">A60</font></i></a>, <a class="ref" href="connsp_1.html#T37" title="CONNSP_1:th.37">CONNSP_1:37</a></i>;<br/><a NAME="E27:105_3_2"/><b>then </b>
<font color="Maroon" title="c22">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">u</font>
 <b>by </b><i><a class="txt" href="#E5:105_3_2"><i><font color="Green" title="E94">A56</font></i></a>, <a class="txt" href="#E8:105_3_2"><i><font color="Green" title="E96">A58</font></i></a>, <a class="txt" href="#E23:105_3_2"><i><font color="Green" title="E102">A64</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">r</font> = <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c22">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="ref" href="toprns_1.html#T29" title="TOPRNS_1:th.29">TOPRNS_1:29</a></i>;<br/><a NAME="E29:105_3_2"/><i><font color="Green" title="E104">A66</font></i>: 
<font color="Maroon" title="c28">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c24">rv</font>
 <b>by </b><i><a class="txt" href="#E24:105_3_2"><i><font color="Green" title="E103">A65</font></i></a>, <a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/><a NAME="E30:105_3_2"/><b>then </b>
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c24">rv</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="jordan.html#T18" target="_self" title="JORDAN:th.18">Th18</a></i>;<br/><a NAME="E31:105_3_2"/><b>then </b><i><font color="Green" title="E105">A67</font></i>: 
<font color="Maroon" title="c23">PC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="txt" href="#E20:105_3_2"><i><font color="Green" title="E101">A63</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E32:105_3_2"/><i><font color="Green" title="E106">A68</font></i>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="jordan.html#T24" target="_self" title="JORDAN:th.24">Th24</a></i>;<br/><a NAME="E33:105_3_2"/>
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>;<br/><a NAME="E34:105_3_2"/><b>then </b><i><font color="Green" title="E107">A69</font></i>: 
<font color="Maroon" title="c11">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="txt" href="#E23:105_3_2"><i><font color="Green" title="E102">A64</font></i></a>, <a class="txt" href="#E32:105_3_2"><i><font color="Green" title="E106">A68</font></i></a>, <a class="ref" href="tops_1.html#T61" title="TOPS_1:th.61">TOPS_1:61</a></i>;<br/><a NAME="E35:105_3_2"/><i><font color="Green" title="E108">A70</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c23">PC</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E36:105_3_2"/><b>then </b><i><font color="Green" title="E109">A71</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="txt" href="#E31:105_3_2"><i><font color="Green" title="E105">A67</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c29">D</font> =  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>;<br/><b>set </b><font color="Maroon" title="c30">S</font> =  <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>;<br/><b>set </b><font color="Maroon" title="c31">B</font> = <span class="p1">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c22">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><a NAME="E37:105_3_2"/><i><font color="Green" title="E110">A72</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="toprealb.html#T9" title="TOPREALB:th.9">TOPREALB:9</a></i>;<br/><a NAME="E38:105_3_2"/><i><font color="Green" title="E111">A73</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="brouwer.html#T3" title="BROUWER:th.3">BROUWER:3</a></i>;<br/><a NAME="E39:105_3_2"/><i><font color="Green" title="E112">A74</font></i>: 
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T17" title="TOPREAL9:th.17">TOPREAL9:17</a></i>;<br/><a NAME="E40:105_3_2"/><i><font color="Green" title="E113">A75</font></i>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19" title="TOPREAL9:th.19">TOPREAL9:19</a></i>;<br/><a NAME="E41:105_3_2"/><i><font color="Green" title="E114">A76</font></i>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T16" title="TOPREAL9:th.16">TOPREAL9:16</a></i>;<br/><a NAME="E42:105_3_2"/>
<font color="Maroon" title="c16">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="txt" href="#E18:105"><i><font color="Green" title="E90">A25</font></i></a>, <a class="txt" href="#E36:105_3_2"><i><font color="Green" title="E109">A71</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E43:105_3_2"/><b>then </b>
<font color="Maroon" title="c16">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E38:105_3_2"><i><font color="Green" title="E111">A73</font></i></a>, <a class="txt" href="#E41:105_3_2"><i><font color="Green" title="E114">A76</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c32">R</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c16">A</font></span>)</span><b> such that </b><br/><a NAME="E45:105_3_2"/><i><font color="Green" title="E115">A77</font></i>: 
<font color="Maroon" title="c32">R</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E46:105_3_2"/><i><font color="Green" title="E116">A78</font></i>: 
<font color="Maroon" title="c32">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c16">A</font>
 <b>by </b><i><a class="txt" href="#E16:105"><i><font color="Green" title="E88">A23</font></i></a>, <a class="txt" href="jordan.html#E89"><i><font color="Green" title="E75">Lm14</font></i></a></i>;<br/><a NAME="E47:105_3_2"/>
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">A</font>
 <b>by </b><i><a class="txt" href="#E16:105"><i><font color="Green" title="E88">A23</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c33">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c22">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E49:105_3_2"/><i><font color="Green" title="E117">A79</font></i>: 
<font color="Maroon" title="c33">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E50:105_3_2"/><i><font color="Green" title="E118">A80</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">P</font> implies <font color="Maroon" title="c33">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <font color="Maroon" title="c33">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="#E10:105"><i><font color="Green" title="E85">A7</font></i></a>, <a class="txt" href="#E17:105"><i><font color="Green" title="E89">A24</font></i></a>, <a class="txt" href="#E18:105"><i><font color="Green" title="E90">A25</font></i></a>, <a class="txt" href="#E1:105_3_2"><i><font color="Green" title="E92">A54</font></i></a>, <a class="txt" href="#E11:105_3_2"><i><font color="Green" title="E98">A60</font></i></a>, <a class="txt" href="#E16:105_3_2"><i><font color="Green" title="E99">A61</font></i></a>, <a class="txt" href="#E34:105_3_2"><i><font color="Green" title="E107">A69</font></i></a>, <a class="txt" href="#E36:105_3_2"><i><font color="Green" title="E109">A71</font></i></a>, <a class="txt" href="#E45:105_3_2"><i><font color="Green" title="E115">A77</font></i></a>, <a class="txt" href="#E46:105_3_2"><i><font color="Green" title="E116">A78</font></i></a>, <a class="txt" href="jordan.html#E91"><i><font color="Green" title="E77">Lm16</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c34">g</font> =  <a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font>;<br/><b>set </b><font color="Maroon" title="c35">h</font> =  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font>;<br/><a NAME="E51:105_3_2"/><i><font color="Green" title="E119">A81</font></i>: 
 <a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="txt" href="#E37:105_3_2"><i><font color="Green" title="E110">A72</font></i></a>, <a class="txt" href="#E38:105_3_2"><i><font color="Green" title="E111">A73</font></i></a>, <a class="txt" href="#E39:105_3_2"><i><font color="Green" title="E112">A74</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">F</font> = <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">f</font></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <b>by </b><i><a class="txt" href="#E37:105_3_2"><i><font color="Green" title="E110">A72</font></i></a>, <a class="txt" href="#E38:105_3_2"><i><font color="Green" title="E111">A73</font></i></a>, <a class="txt" href="#E39:105_3_2"><i><font color="Green" title="E112">A74</font></i></a>, <a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/><a NAME="E53:105_3_2"/>
<font color="Maroon" title="c22">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T17" target="_self" title="JORDAN:th.17">Th17</a></i>;<br/><a NAME="E54:105_3_2"/><b>then </b>
 <a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T64" target="_self" title="JORDAN:th.64">Th64</a></i>;<br/><a NAME="E55:105_3_2"/><b>then </b><i><font color="Green" title="E120">A82</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E49:105_3_2"><i><font color="Green" title="E117">A79</font></i></a></i>;<br/><a NAME="E56:105_3_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">p</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c22">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="toprns_1.html#T29" title="TOPRNS_1:th.29">TOPRNS_1:29</a></i>;<br/><a NAME="E57:105_3_2"/><b>then </b><i><font color="Green" title="E121">A83</font></i>: 
<font color="Maroon" title="c22">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/><a NAME="E58:105_3_2"/><b>then </b>
 <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan.html#T66" target="_self" title="JORDAN:th.66">Th66</a></i>;<br/><a NAME="E59:105_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">f</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E55:105_3_2"><i><font color="Green" title="E120">A82</font></i></a></i>;<br/><a NAME="E60:105_3_2"/><b>then </b><i><font color="Green" title="E122">A84</font></i>: 
<font color="Maroon" title="c36">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="#E51:105_3_2"><i><font color="Green" title="E119">A81</font></i></a>, <a class="ref" href="pre_topc.html#T56" title="PRE_TOPC:th.56">PRE_TOPC:56</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c37">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c36">F</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_3_2_1_1"/>
( <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c36">F</font> or  not <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c36">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:105_3_2_1_1_1"/><i><font color="Green" title="E123">A85</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c36">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c36">F</font></span><br/><div class="add"><a NAME="E2:105_3_2_1_1_1"/><i><font color="Green" title="E124">A86</font></i>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K2" title="TOPREAL9:func.2">cl_Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T18" title="TOPREAL9:th.18">TOPREAL9:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_3_2_1_1_1_1_1"/>
( <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> or <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> )
 </span><b>by </b><i><a class="txt" href="#E38:105_3_2"><i><font color="Green" title="E111">A73</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1"><i><font color="Green" title="E123">A85</font></i></a>, <a class="txt" href="#E2:105_3_2_1_1_1"><i><font color="Green" title="E124">A86</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_3_2_1_1_1_1_1_1"/><i><font color="Green" title="E125">A87</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c36">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">x</font></span><br/><div class="add"><a NAME="E2:105_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c36">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:105_3_2_1_1_1"><i><font color="Green" title="E123">A85</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E3:105_3_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c36">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">x</font>
 <b>by </b><i><a class="txt" href="#E37:105_3_2"><i><font color="Green" title="E110">A72</font></i></a>, <a class="txt" href="#E40:105_3_2"><i><font color="Green" title="E113">A75</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1_1_1_1"><i><font color="Green" title="E125">A87</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_3_2_1_1_1_1_1_2"/><i><font color="Green" title="E125">A88</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c36">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">x</font></span><br/><div class="add"><a NAME="E2:105_3_2_1_1_1_1_1_2"/><i><font color="Green" title="E126">A89</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c33">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:105_3_2_1_1_1_1_1_2"/><i><font color="Green" title="E127">A90</font></i>: 
<font color="Maroon" title="c11">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><b>set </b><font color="Maroon" title="c38">SS</font> =  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>;<br/><a NAME="E4:105_3_2_1_1_1_1_1_2"/>
 <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_2_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c39">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> or <font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<b>assume </b><a NAME="E1:105_3_2_1_1_1_1_1_2_1"/><i><font color="Green" title="E128">A91</font></i>: 
<font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<b>assume </b><a NAME="E2:105_3_2_1_1_1_1_1_2_1"/>
not <font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:105_3_2_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E129">A92</font></i>: 
<font color="Maroon" title="c39">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="#E1:105_3_2_1_1_1_1_1_2_1"><i><font color="Green" title="E128">A91</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c40">a</font> = <font color="Maroon" title="c39">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E1:105_3_2_1_1_1_1_1_2_1"><i><font color="Green" title="E128">A91</font></i></a></i>;<br/>
<a NAME="E5:105_3_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c40">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">PC</font>
 
<b>by </b><i><a class="txt" href="#E35:105_3_2"><i><font color="Green" title="E108">A70</font></i></a>, <a class="txt" href="#E3:105_3_2_1_1_1_1_1_2_1"><i><font color="Green" title="E129">A92</font></i></a></i>;<br/>
<a NAME="E6:105_3_2_1_1_1_1_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c40">a</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c22">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">rv</font>
 
<b>by </b><i><a class="txt" href="#E20:105_3_2"><i><font color="Green" title="E101">A63</font></i></a>, <a class="ref" href="topreal9.html#T7" title="TOPREAL9:th.7">TOPREAL9:7</a></i>;<br/>
<b>hence </b><a NAME="E7:105_3_2_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E29:105_3_2"><i><font color="Green" title="E104">A66</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1_1_1_2_1"><i><font color="Green" title="E128">A91</font></i></a>, <a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c39">SS</font> =  <a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E6:105_3_2_1_1_1_1_1_2"/><i><font color="Green" title="E128">A93</font></i>: 
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c39">SS</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T9" title="TOPREAL9:th.9">TOPREAL9:9</a></i>;<br/><a NAME="E7:105_3_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c39">SS</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/><a NAME="E8:105_3_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c39">SS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">U</font> or <font color="Maroon" title="c39">SS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">U</font> )
 <b>by </b><i><a class="txt" href="#E3:105"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></i>;<br/><a NAME="E9:105_3_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E129">A94</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">P</font>
 <b>by </b><i><a class="txt" href="#E2:105"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="#E23:105_3_2"><i><font color="Green" title="E102">A64</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E125">A88</font></i></a>, <a class="txt" href="#E6:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E128">A93</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E10:105_3_2_1_1_1_1_1_2"/><i><font color="Green" title="E130">A95</font></i>: 
<span class="p1">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E57:105_3_2"><i><font color="Green" title="E121">A83</font></i></a>, <a class="ref" href="jordan.html#T65" target="_self" title="JORDAN:th.65">Th65</a></i>;<br/><a NAME="E11:105_3_2_1_1_1_1_1_2"/>
 <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font> <a href="abian.html#NR4" title="ABIAN:NR.4">has_no_fixpoint</a> 
 <b>by </b><i><a class="txt" href="#E57:105_3_2"><i><font color="Green" title="E121">A83</font></i></a>, <a class="ref" href="jordan.html#T67" target="_self" title="JORDAN:th.67">Th67</a></i>;<br/><a NAME="E12:105_3_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E131">A96</font></i>: 
not <font color="Maroon" title="c37">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font>
 <b>by </b><i><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/><a NAME="E13:105_3_2_1_1_1_1_1_2"/><i><font color="Green" title="E132">A97</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" title="TOPREALB:func.7">Tcircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><font color="Maroon" title="c36">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> = 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:105_3_2_1_1_1"><i><font color="Green" title="E123">A85</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c33">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E38:105_3_2"><i><font color="Green" title="E111">A73</font></i></a>, <a class="txt" href="#E39:105_3_2"><i><font color="Green" title="E112">A74</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E125">A88</font></i></a>, <a class="txt" href="#E2:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E126">A89</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan.html#K9" title="JORDAN:func.9">DiskProj</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E38:105_3_2"><i><font color="Green" title="E111">A73</font></i></a>, <a class="txt" href="#E39:105_3_2"><i><font color="Green" title="E112">A74</font></i></a>, <a class="txt" href="#E50:105_3_2"><i><font color="Green" title="E118">A80</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E125">A88</font></i></a>, <a class="txt" href="#E3:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E127">A90</font></i></a>, <a class="txt" href="#E9:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E129">A94</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="topreal9.html#K3" title="TOPREAL9:func.3">Sphere</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E125">A88</font></i></a>, <a class="txt" href="#E10:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E130">A95</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan.html#K10" title="JORDAN:func.10">RotateCircle</a> <font color="Maroon" title="c22">p</font>,<font color="Maroon" title="c28">r</font>,<font color="Maroon" title="c22">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font>
<b>by </b><i><a class="txt" href="#E1:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E125">A88</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E15:105_3_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c36">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c37">x</font>
 <b>by </b><i><a class="txt" href="#E37:105_3_2"><i><font color="Green" title="E110">A72</font></i></a>, <a class="txt" href="#E1:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E125">A88</font></i></a>, <a class="txt" href="#E12:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E131">A96</font></i></a>, <a class="txt" href="#E13:105_3_2_1_1_1_1_1_2"><i><font color="Green" title="E132">A97</font></i></a>, <a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:105_3_2_1_1_1"/>
not <font color="Maroon" title="c37">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c36">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:105_3_2_1_1_2"/>
not <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c36">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Olive">b<sub>1</sub></font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c36">F</font></span><br/><div class="add"><b>hence </b><a NAME="E2:105_3_2_1_1_2"/>
not <font color="Maroon" title="c37">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a> <font color="Maroon" title="c36">F</font>
 <b>by </b><i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E62:105_3_2"/><b>then </b>
not <font color="Maroon" title="c36">F</font> <a href="abian.html#R3" title="ABIAN:pred.3">has_a_fixpoint</a> 
 <b>by </b><i><a class="ref" href="abian.html#D5" title="ABIAN:def.5">ABIAN:def 5</a></i>;<br/><b>hence </b><a NAME="E63:105_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E60:105_3_2"><i><font color="Green" title="E122">A84</font></i></a>, <a class="ref" href="brouwer.html#T14" title="BROUWER:th.14">BROUWER:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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